A certain stock has earned 14%, 8.57%, -12.1%, 2.1%, and 9.43% over the past 5 years. 
Construct a confidence interval with 95% probability for the range of options for this stock

To construct a 95% confidence interval for the average return of the stock over the past 5 years, we follow these steps:

1. **Calculate the sample mean (average) return**:
   \[
   \bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i
   \]
   where \( x_i \) are the stock returns and \( n \) is the number of years.

2. **Calculate the sample standard deviation**:
   \[
   s = \sqrt{\frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x})^2}
   \]
   
3. **Determine the standard error of the mean**:
   \[
   SE = \frac{s}{\sqrt{n}}
   \]

4. **Find the t-value for a 95% confidence level** for \( n-1 \) degrees of freedom from the t-distribution.

5. **Construct the confidence interval**:
   \[
   \text{CI} = \bar{x} \pm (t \times SE)
   \]

Let's go through these calculations step-by-step.

### Step 1: Calculate the sample mean
The stock returns over the past 5 years are:
14%, 8.57%, -12.1%, 2.1%, and 9.43%.

\[
\bar{x} = \frac{14 + 8.57 + (-12.1) + 2.1 + 9.43}{5}
\]

\[
\bar{x} = \frac{22}{5} = 4.4\%
\]

### Step 2: Calculate the sample standard deviation

Now, we compute the deviations from the mean and square them:

| Year  | Return (%) | Deviation from Mean (\( x_i - \bar{x} \)) | Squared Deviation \((x_i - \bar{x})^2\) |
|-------|------------|--------------------------------------------|----------------------------------------|
| 1     | 14         | \( 14 - 4.4 = 9.6 \)                      | \( 9.6^2 = 92.16 \)                   |
| 2     | 8.57       | \( 8.57 - 4.4 = 4.17 \)                   | \( 4.17^2 = 17.39 \)                  |
| 3     | -12.1      | \( -12.1 - 4.4 = -16.5 \)                 | \( (-16.5)^2 = 272.25 \)              |
| 4     | 2.1        | \( 2.1 - 4.4 = -2.3 \)                    | \( (-2.3)^2 = 5.29 \)                 |
| 5     | 9.43       | \( 9.43 - 4.4 = 5.03 \)                   | \( 5.03^2 = 25.3 \)                   |

The sum of the squared deviations is:
\[
92.16 + 17.39 + 272.25 + 5.29 + 25.3 = 412.39
\]

Now, the sample variance is:
\[
s^2 = \frac{412.39}{5 - 1} = \frac{412.39}{4} = 103.0975
\]

The sample standard deviation is:
\[
s = \sqrt{103.0975} \approx 10.15\%
\]

### Step 3: Calculate the standard error
The standard error is:
\[
SE = \frac{s}{\sqrt{n}} = \frac{10.15}{\sqrt{5}} \approx \frac{10.15}{2.236} \approx 4.54\%
\]

### Step 4: Find the t-value
For a 95% confidence interval and 4 degrees of freedom (\(n-1 = 5-1 = 4\)), the critical t-value from the t-distribution is approximately \( t = 2.776 \).

### Step 5: Construct the confidence interval
The confidence interval is given by:
\[
\text{CI} = \bar{x} \pm (t \times SE) = 4.4\% \pm (2.776 \times 4.54\%)
\]

\[
\text{CI} = 4.4\% \pm 12.6\%
\]

Thus, the 95% confidence interval for the stock's average return is approximately:
\[
[-8.2\%, 16.99\%]
\]

This means we are 95% confident that the true average return of the stock lies between -8.2% and 16.99%.

---

Would you like further details on any of the steps? Here are 5 related questions you might find interesting:

1. How do we interpret confidence intervals in financial returns?
2. What assumptions are made in constructing this confidence interval?
3. How would the confidence interval change with more data points?
4. How does the t-distribution differ from the normal distribution in this context?
5. Can we use this confidence interval to predict future stock performance?

**Tip**: A wider confidence interval indicates greater uncertainty about the stock's average return. Reducing variability or increasing sample size can narrow the interval.

A certain stock has earned 14%, 8.57%, -12.1%, 2.1%, and 9.43% over the past 5 years. Construct a confidence interval with 95% probability for the range of options for this stock.

Construct a 95% confidence interval for stock returns over 5 years with returns of 14%, 8.57%, -12.1%, 2.1%, and 9.43%. Learn about sample mean, standard deviation, and how to calculate the confidence interval using t-distribution. Suitable for university-level statistics or finance students.

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